Why must a preference relation for an agent be over a set of alternatives for which they can choose from instead of any set?

Question

A preference relation is of course a reflexive, total and transitive binary relation on a set, with the additional requirement that the agent be able to select one alternative from the set. I don't understand why this last requirement is needed?

In particular, I once heard from someone I consider much smarter than myself that for a player i in a normal form game, it doesn't really make sense to compare u_i(s) and u_i(t) for arbitrary strategy profiles s, t because player i can't necessarily deviate from s to t and vice versa.

But why wouldn't it make sense for players to assign preferences to the possible outcomes of a game, even though they don't directly get to choose amongst the outcomes, only contribute to it?

Furthermore:

1. If u_i(s) and u_i(t) are not in general comparable, how does it make sense to take linear combinations of these for expected utilities?

2. Notions of equivalence then lose preservation of players preferences over Nash equilibria, which is relevant when considering equilibrium selection.

Answer 1

A preference relation is just a relation on the set of choices. I think the Wikipedia explanation "among which a consumer can choose" was meant to clarify what the set represents not impose a formal restriction.

A rational preference relation, however, is also total and transitive. Note totality on $(x,x)$ implies $x \succeq x$ or $x \succeq x$, which is reflexivity.

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I'm not sure about the remark about comparison. While it is true that you cannot deviate from one strategy profile to another mid-game (because your strategy profile completely specifies what to do after any combination of moves your opponent produces), you can still choose among different strategy profiles and mix them stochastically.

For example, a strategy profile $q$ could be defined that selects $s$ with probability $0.6$ and $t$ with probability $0.4$. This choice happens pre-game and requires no mid-game deviations. Then $$u_i(q)=0.6u_i(s)+0.4u_i(t).$$ Perhaps a strategy profile can be thought of as a way of reducing a sequential game to a single-move game where the payoff is the utility of the whole strategy profile.

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I'm not sure I understand the question 2, perhaps you could clarify?

Answer 2

The maximal elements of a preference relation are supposed to be the acts among which the agent will choose. If you add in a whole bunch of impossible options, it might well be that none of the maximal elements are feasible.

You can define a preference relation over a set that includes options like "Defect if and only if my opponent defects" and you can prefer them to the other options. But unless you can perform that action, then it is irrelevant. So I guess there's no theoretically strong ground for the restriction, but it seems reasonable in practice.